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Consider the hyperbola H: x^(2)-y^(2)=1 ...

Consider the hyperbola `H: x^(2)-y^(2)=1` and a circle S with centre `N(x_(2),0)`. Suppose that H ans S touch each other at a point `P(x_(1),y_(1))` with `x_(1)gt1` and `y_(1)gt0`. The common tangent to H ans S at P intersects the x-axis at point M. If (l, m) is the centroid of the triangle PMN, then the correct expression(s) is (are)-

A

`(dl)/(dx_(1))=1-(1)/(3x_(1)^(2))" for "x_(1)gt1`

B

`(dm)/(dx_(1))=(x_(1))/(3(sqrt(x_(1)^(2)-1)))" for "x_(1)gt1`

C

`(dl)/(dx_(1))=1+(1)/(3x_(1)^(2))" for "x_(1) gt1`

D

`(dm)/(dy_(1))=(1)/(3)" for "y_(1)gt0`

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